Ramsey Number of K_(2,s 1) vs.K_(1,n)

It is shown that the Ramsey number r(K_(2,s 1),K_(1,n))(?)n sn~(1/2) (s 3)/2 o(1) for large n,and r(K_(2,s 1),K_(1,n))∈{((q-1)~2/s) 1,((q-1)~2/s) 2},where n=((q-1)~2/s)-q 2 and q is a prime power such that s|(q-1).     

Regularity for weakly (K1, K2)-quasiregular mappings

In this paper, we first give the definition of weakly (K1, K2)-quasiregular mappings, and then by using the Hodge decomposition and the weakly reverse Holder inequality, we obtain their regularity property: For any ql that satisfies 0 < K1n(n+4)/22n+1 × 100n2[23n/2(25n + 1)](n - q1) < 1, there exists p1 = p1(n, q1, K1, K2) > n, such that any (K1, K2)-quasiregular mapping f ∈W(loc)(1,q1)(Ω,Rn) is in fact in W(loc)(1,p1)(Ω,Rn). That is, f is (K1, K2)-quasiregular in the usual sense.     

The Einstein-Kähler metric on the third Cartan-Hartogs domain

In this paper we discuss the Einstein-Kahler metric on the third Cartan-Hartogs domain Y111（n, q; K）. Firstly we get the complete Einstein Kahler metric with explicit form on Y111（n, q; K） in the case of K=q/2 ＋ 1/q-1. Secondly we obtain the holomorphic sectional curvature under this metric and get the sharp estimate for this holomorphic curvature. Finally we prove that the complete Einstein-Kahler metric is equivalent to the Bergman metric on Y111（n, q; K） in case of K=q/2＋1/q-1.     

Ramsey Number r(C_(2m 1), K_n) With Large n

This is an announcement that r(C2m 1, Kn) ＜ c(m) ( ) 1/m has been proved.The Ramsey number r(H, Kn) is the smallest integer N such that every H-free graph onN vertices has independence number at least n. The study of Ramsey number r(Ck, Kn) wasinitiated by Bondy and Erd s[2]. They proved that for any fixed n, r(Ck, Kn) = (k - 1)(n - 1) 1if k n2 - 1, and r(Ck, Kn) kn2. For fixed k 3, it is difficult to obtain a satisfied bound ofr(Ck, Kn) for n → ∞ . The bound of Bondy and Erd s w…     

分数次型 Marcinkiewicz 积分在 Hardy 空间中的有界性

The authors in the paper proved that if Ω is homogeneous of degree zero and satisfies some certain logarithmic type Lipschitz condition,then the fractional type Marcinkiewicz Integral μ Ω,α is an operator of type （H˙ K n（1-1/q 1 ）,p q 1 ,˙ K n（1-1/q 1 ）,p q 2 ） and of type （H 1 （R n ）,L n/（n-α） ）.     

A Combinatorial Theorem on Ordered Circular Sequences of n_1 u''''s and n_2 v''''s with Application to Kernel-perfect Graphs

Abstract An ordered circular permutation S of u's and v's is called an ordered circular sequence of u's andv's.A kernel of a digraph G=(V,A)is an independent subset of V,say K,such that for any vertex v_i in V\Kthere is an arc from v_i to a vertex v_j in K.G is said to be kernel-perfect(KP)if every induced subgraph of Ghas a kernel. G is said to be kernel-perfect-critical(KPC)if G has no kernel but every proper induced subgraphof G has a kernel.The digraph G=(V,A)=(j_1,j_2,…,j_k)is defined by:V(G)={0,1,…,n-1},A(G)={uv│v-u≡j_i(mod n) for 1≤i≤k}. In an eariler work, we investigated the digraph G=(1,±δd,±2d,±3d,…±sd),denoted by G(n,d,r,s),whereδ=1 for d>1 or δ=0 for d=1,and n,d,r,s are positive integers with(n,d)=r and n=mr ,and gave some necessaryand sufficient conditions for G(n,d,r,s)with r≥3 and s=1 to be KP or KPC. In this paper,we prove a combinatorial theorem on ordered circular sequences of n_1 u's and n_2 v's.By usingthe theorem,we prove that,if(n,d)=r≥2 and s≥2,then G(n,d,r,s,)is     

On K_(1,k)-factorization of bipartite multigraphs

A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n,which partition the set of edges of λKm,n.In this paper,it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n,whenever k is any positive integer,is that(1) m ≤ kn,(2) n ≤ km,(3) km-n ≡ kn-m ≡ 0(mod(k2-1)) and(4) λ(km-n)(kn-m) ≡ 0(mod k(k -1)(k2 -1)(m n)).     

On the Convergence of Products γ^-sh1hn in the Adams Spectral Sequence

Abstract Let A be the mod p Steenrod algebra and S the sphere spectrum localized at p, where p is an odd prime. In 2001 Lin detected a new family in the stable homotopy of spheres which is represented by （b0hn-h1bn-1）∈ ExtA^3,（p^n＋p）q（Zp,Zp） in the Adams spectral sequence. At the same time, he proved that i.（hlhn） ∈ExtA^2,（p^n＋P）q（H^＊M, Zp） is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element ξn∈π（p^n＋p）q-2M. In this paper, with Lin＇s results, we make use of the Adams spectral sequence and the May spectral sequence to detect a new nontrivial family of homotopy elements jj′j^-γsi^-i′ξn in the stable homotopy groups of spheres. The new one is of degree p^nq ＋ sp^2q ＋ spq ＋ （s - 2）q ＋ s - 6 and is represented up to a nonzero scalar by hlhnγ-s in the E2^s＋2,＊-term of the Adams spectral sequence, where p ≥ 7, q = 2（p - 1）, n ≥ 4 and 3 ≤ s 〈 p.     

LIE ALGEBRA K(n,μj,m)OF CARTAN TYPE OF CHARACTERISTIC p=2.

Let K(n,μ_j,m),n=2r+1,denote the Lie algebra of characteristic p=2,which isdefined in [4].In the paper the restrietability of K(n,μ_j,m)is discussed and it is provedthat,when r≡1(mod 2)and r>1,1(ad.f)=n+1 if and only if 0≠f∈.Then theinvariance of some filtrations of K(n,μ_j,m)and the condition of isomorphism of K(n,μ_j,m)and K(n′,μ_j′,m′)are obtained.Besides,the generators and the derivation algebraof K(n,μ_l,m)are discussed.The results also hold,when r=0(mod 2)and r>0.     

关于特定同余式方程解数的余项（英文）

Let f(x) be an irreducible polynomial of degree m ≥ 2 with integer coefficients,and let r(n) denote the number of solutions x of the congruence f(x) ≡ 0(mod n) satisfying0 ≤ x n. Define ?(x) =Σ 1≤n≤x r(n)-αx, where α is the residue of the Dedekind zeta function ζ(s, K) at its simple pole s = 1. In this paper it is shown that ∫_1~X?~2(x)dx? ε{X~(3-6/m+3+ε)if m ≥ 3,X~(2+ε) if m = 2,for any non-Abelian polynomial f(x) and any ε 0. This result constitutes an improvement upon that of Lü for the error terms on average.     

K1,m∨Pn的均匀全染色

对一个正常的全染色满足各种颜色所染元素数(点或边)相关不超过1时,称为均匀全染色,其所用最少染色数称为均匀全色数.就星K1,m与路Pn的联图K1,m∨Pn,得到了在m,n不同取值情况下的均匀全色数.     

完全二部图K_(1,n),K_(2,n)和K_(3,n)的点强可区别全染色

设f是图G的一个正常全染色.对任意x∈V(G),令C(x)表示与点x相关联或相邻的元素的颜色以及点x的颜色所构成的集合.若对任意u,v∈V(G),u≠v,有C(u)≠C(v),则称.f是图G的一个点强可区别全染色,对一个图G进行点强可区别全染色所需的最少的颜色的数目称为G的点强可区别全色数,记为X_(vst)(G).讨论了完全二部图K_(1,n),K_(2,n)和L_(3,n)的点强可区别全色数,利用组合分析法,得到了当n≥3时,X_(vst)(K_(1,n)=n+1,当n≥4时,X_(vst)(K_(2,n)=n+2,当n≥5时,X_(vst)(K_(3,n))=n+2.     

Linear Chromatic Bounds for a Subfamily of 3K 1-free Graphs

For an arbitrary class of graphs , there may not exist a function f such that , for every . When such a function exists, it is called a χ-binding function for . The problem of finding an optimal χ-binding function for the class of 3K ₁-free graphs is open. In this paper, we obtain linear χ-binding function for the class of {3K ₁, H}-free graphs, where H is one of the following graphs: , House graph and Kite graph. We first describe structures of these graphs and then derive χ-binding functions.     

K3 surfaces,Picard numbers and Siegel disks

If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between 0 and 18. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type fixed point formulas and related Grothendieck residues.     

Off‐diagonal and asymptotic results on the Ramsey number r(K2,m,K2,n)

An upper bound on the Ramsey number r(K_2,n‐s,K_2,n) where s ≥ 2 is presented. Considering certain r(K_2,n‐s,K_2,n)‐colorings obtained from strongly regular graphs, we additionally prove that this bound matches the exact value of r(K_2,n‐s,K_2,n) in infinitely many cases if holds. Moreover, the asymptotic behavior of r(K_2,m,K_2,n) is studied for n being sufficiently large depending on m. We conclude with a table of all known Ramsey numbers r(K_2,m,K_2,n) where m,n ≤ 10. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 252–268, 2003     

On blocks with K(B)-L(B) = 1

We use the method of local representation and original method of Brauer to study the block with K(B) - L(B) = 1, and get some properties on the defect group and the structure of this kind of blocks. Then, we show that K(B) conjecture holds for this kind of blocks.     

K1,4-受限图的路可扩性

图G中同构于K1,p的子图叫G的p-爪(p3).如果G中任意一个p-爪中1度顶点之间边的数目p-2,则称G为K1,p-受限图,它是无爪图(p=3时)的推广.本文证明了:连通、局部3-连通的K1,4-受限图是路可扩的.     

K1,p-受限图

图G中同构于Ki,p的子图叫G的p-爪(P≥3)．如果G中任意一个p-爪中1度顶点之间边(在G中的边)的数目≥P-2,则称G为K1,p-受限图,它是无爪图的推广．本文证明了连通、局部2-连通的K1,4-受限图是完全圈可扩的．     

On three color Ramsey numbers R(C4,C4,K1,n)

For $k$ given graphs $G_1,G_2,\dots ,G_k$, $k\ge 2$, the $k$-color Ramsey number, denoted by $R(G_1,G_2,\dots ,G_k)$, is the smallest integer $N$ such that if we arbitrarily color the edges of a complete graph of order $N$ with $k$ colors, then it always contains a monochromatic copy of $G_i$ colored with $i$, for some $1\le i\le k$. Let $C_m$ be a cycle of length $m$ and $K_{1,n}$ a star of order $n+1$. In this paper, firstly we give a general upper bound of $R(C_4,C_4,\dots ,C_4,K_{1,n})$. In particular, for the 3-color case, we have $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+3$ and this bound is tight in some sense. Furthermore, we prove that $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+2$ for all $n={?}^2??$ and $?\ge 2$, and if $?$ is a prime power, then the equality holds.